(12) Why did the Levites not receive a portion in the inheritance of Eretz Yisrael and in the spoils…? Because they were set aside to serve G-d… and to teach Torah: "They will teach Your… Torah to Israel [Devarim 33, 10}"… Therefore they were set apart from the ways of the world. They do not wage war like the rest of Israel, nor do they receive an inheritance, nor do they acquire for themselves through their physical power. Instead, they are G-d's legion, as it [ibid.:11]: states: "G-d has blessed His legion" and He provides for them [the tithes], as it [Bamidbar 18, 20] states: "I am your portion and your inheritance."
(13) Not only the tribe of Levi, but any inhabitant of the world
motivated… to set himself aside and stand before G-d to serve and minister to Him and to know G-d… removing from his neck the yoke of the [physical world], he is sanctified… G-d will be his portion and heritage forever and will provide what is sufficient for him in this world like He provides for the priests and the Levites. And thus David declared: "G-d is the lot of my portion, my goblet, You support my lot (Tehilim 16, 5)."

If the Rambam exempted yeshiva boys, he would have said so in the laws of war (precisely called "The Laws of Kings and Their Wars"), together with all other exemptions. It immediately sounds suspicious that this idea is found among the agricultural laws of Sh'mitta! The obvious reason is that this is the final paragraph in the Rambam's extensive Book of Agricultural Laws. The Rambam (copying the Talmud) finishes his Books with nice ideas, in place of the regular halachot. Here too, after an entire Book teaching we must give the various tithes mainly to the Kohen and Levi, many regular Jews will ask, "Why am I so less fortunate or spiritual, just because I'm not from Levi?" Comes the Rambam to appease, that we can also "be spiritual like Levi". Nevertheless, this obviously isn't meant to be taken halachically, for surely the Rambam forbids a yeshiva boy to eat Bikurim, Trumah or Hallah as if he were a Kohen. Similarly, the seeming exemption from war is also "biological", and seriously forbidden to be taken literally nor halachically.
b. The undisputed Mishna (Sotah 8, 7) & the Rambam himself write to the contrary ["in Milchemet Mitzvah (a defensive war) everyone goes, even a groom from the yichud room…", Hil. Laws of Kings and Wars, 7, 4), and even on Shabbat (2, 23, and Shulchan Aruch, Or. Ch. 329, 6-9, "it's a mitzvah for ALL ISRAEL whoever can come and aid their brothers under siege and save them from the gentiles"). the explicit halacha is that in a defensive war, even on Shabbat, it is a mitzvah upon all Israel to come and fight to defend their brothers and save them from the gentiles. In fact, the Rambam writes there that rabbis and learned Jews are supposed to be the first ones to do so. Any other understanding contradicts the entire Tanach, where ALL men are counted for war (Bamidbar 1), including our spiritual heroes, for there's no "religious" exemptions from Milchamot Mitzva.
c. Even if one tries to take this literally, and thinks that any yeshiva boy can volunteer to be just like Levi, we know that Kohanim and Levi'im DO (!) go to Milchemet Mitzvah like the rest of Israel. They may be exempt from optional wars, but everyone knows (from "Al HaNissim") the MACCABEES were KOHANIM, as was Benayahu ben Yehoyada a general of King David's (Chronicles I 27, 5), and regarding "the enemies of G-d where it's a mitzvah [to fight]… against the nations who trouble Israel, there's no differentiation between Kohanim and Israel…" (Radak, Shmuel II, 23, 20). Accordingly, "they do not wage war like the rest of Israel", may exempt them from OPTIONAL wars (offensive, to conquer foreign lands), but not from defensive Milchemet Mitzvah.
Even R. Velvel Soloveichik & the Chazon Ish (Eruvin, Likutim 114, 3), the classic Haredi poskim (!), prove that the Rambam means to say here that even Kohanim and Levites themselves are only exempt if they are not necessary [on Sotah 43a, citing the Rambam 7, 8, that a Kohen who takes a new wife who is prohibited for him (e.g. a divorcee) is NOT exempt even from optional wars]. But if they are needed, as is the case today, they must serve even in optional (!) wars, even giving many of their lives just to help the Jewish national pride or economy, and how much more so, in obligatory wars.
d. How can one prove there an exemption for yeshiva boys from King David (among the greatest warriors in Jewish history), whom the Rambam cites there. To the contrary, when David proclaims "G-d is the lot of my portion", he is including going to the wars of mitzvah, which is part of serving G-d, even more so than observing Shabbat (which it "supersedes").